Internal problem ID [10900]
Internal file name [OUTPUT/10157_Sunday_December_31_2023_11_03_01_AM_84918379/index.tex]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-3 Equation of form
\((a x + b)y''+f(x)y'+g(x)y=0\)
Problem number: 77.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_ode_missing_y"
Maple gives the following as the ode type
[[_2nd_order, _missing_y]]
\[ \boxed {x y^{\prime \prime }+\left (a x +b \right ) y^{\prime }=-c x \left (-c \,x^{2}+a x +b +1\right )} \]
This is second order ode with missing dependent variable \(y\). Let \begin {align*} p(x) &= y^{\prime } \end {align*}
Then \begin {align*} p'(x) &= y^{\prime \prime } \end {align*}
Hence the ode becomes \begin {align*} x p^{\prime }\left (x \right )+\left (a x +b \right ) p \left (x \right )+c x \left (-c \,x^{2}+a x +b +1\right ) = 0 \end {align*}
Which is now solve for \(p(x)\) as first order ode.
Entering Linear first order ODE solver. In canonical form a linear first order is \begin {align*} p^{\prime }\left (x \right ) + p(x)p \left (x \right ) &= q(x) \end {align*}
Where here \begin {align*} p(x) &=\frac {a x +b}{x}\\ q(x) &=-\frac {-c^{2} x^{3}+a c \,x^{2}+b c x +x c}{x} \end {align*}
Hence the ode is \begin {align*} p^{\prime }\left (x \right )+\frac {\left (a x +b \right ) p \left (x \right )}{x} = -\frac {-c^{2} x^{3}+a c \,x^{2}+b c x +x c}{x} \end {align*}
The integrating factor \(\mu \) is \begin{align*} \mu &= {\mathrm e}^{\int \frac {a x +b}{x}d x} \\ &= {\mathrm e}^{a x +b \ln \left (x \right )} \\ \end{align*} Which simplifies to \[ \mu = x^{b} {\mathrm e}^{a x} \] The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}}\left ( \mu p\right ) &= \left (\mu \right ) \left (-\frac {-c^{2} x^{3}+a c \,x^{2}+b c x +x c}{x}\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left (x^{b} {\mathrm e}^{a x} p\right ) &= \left (x^{b} {\mathrm e}^{a x}\right ) \left (-\frac {-c^{2} x^{3}+a c \,x^{2}+b c x +x c}{x}\right )\\ \mathrm {d} \left (x^{b} {\mathrm e}^{a x} p\right ) &= \left (-x^{b} {\mathrm e}^{a x} c \left (-c \,x^{2}+a x +b +1\right )\right )\, \mathrm {d} x \end {align*}
Integrating gives \begin {align*} x^{b} {\mathrm e}^{a x} p &= \int {-x^{b} {\mathrm e}^{a x} c \left (-c \,x^{2}+a x +b +1\right )\,\mathrm {d} x}\\ x^{b} {\mathrm e}^{a x} p &= -\frac {\left (-a \right )^{-b} c^{2} \left (x^{b} \left (-a \right )^{b} b \left (b^{2}+3 b +2\right ) \Gamma \left (b \right ) \left (-a x \right )^{-b}-x^{b} \left (-a \right )^{b} \left (a^{2} x^{2}-a b x -2 a x +b^{2}+3 b +2\right ) {\mathrm e}^{a x}-x^{b} \left (-a \right )^{b} b \left (b^{2}+3 b +2\right ) \left (-a x \right )^{-b} \Gamma \left (b , -a x \right )\right )}{a^{3}}-\frac {\left (-a \right )^{-b} c \left (x^{b} \left (-a \right )^{b} \left (1+b \right ) b \Gamma \left (b \right ) \left (-a x \right )^{-b}+x^{b} \left (-a \right )^{b} \left (a x -b -1\right ) {\mathrm e}^{a x}-x^{b} \left (-a \right )^{b} \left (1+b \right ) b \left (-a x \right )^{-b} \Gamma \left (b , -a x \right )\right )}{a}+\frac {\left (-a \right )^{-b} c b \left (x^{b} \left (-a \right )^{b} b \Gamma \left (b \right ) \left (-a x \right )^{-b}-x^{b} \left (-a \right )^{b} {\mathrm e}^{a x}-x^{b} \left (-a \right )^{b} b \left (-a x \right )^{-b} \Gamma \left (b , -a x \right )\right )}{a}+\frac {\left (-a \right )^{-b} c \left (x^{b} \left (-a \right )^{b} b \Gamma \left (b \right ) \left (-a x \right )^{-b}-x^{b} \left (-a \right )^{b} {\mathrm e}^{a x}-x^{b} \left (-a \right )^{b} b \left (-a x \right )^{-b} \Gamma \left (b , -a x \right )\right )}{a} + c_{1} \end {align*}
Dividing both sides by the integrating factor \(\mu =x^{b} {\mathrm e}^{a x}\) results in \begin {align*} p \left (x \right ) &= x^{-b} {\mathrm e}^{-a x} \left (-\frac {\left (-a \right )^{-b} c^{2} \left (x^{b} \left (-a \right )^{b} b \left (b^{2}+3 b +2\right ) \Gamma \left (b \right ) \left (-a x \right )^{-b}-x^{b} \left (-a \right )^{b} \left (a^{2} x^{2}-a b x -2 a x +b^{2}+3 b +2\right ) {\mathrm e}^{a x}-x^{b} \left (-a \right )^{b} b \left (b^{2}+3 b +2\right ) \left (-a x \right )^{-b} \Gamma \left (b , -a x \right )\right )}{a^{3}}-\frac {\left (-a \right )^{-b} c \left (x^{b} \left (-a \right )^{b} \left (1+b \right ) b \Gamma \left (b \right ) \left (-a x \right )^{-b}+x^{b} \left (-a \right )^{b} \left (a x -b -1\right ) {\mathrm e}^{a x}-x^{b} \left (-a \right )^{b} \left (1+b \right ) b \left (-a x \right )^{-b} \Gamma \left (b , -a x \right )\right )}{a}+\frac {\left (-a \right )^{-b} c b \left (x^{b} \left (-a \right )^{b} b \Gamma \left (b \right ) \left (-a x \right )^{-b}-x^{b} \left (-a \right )^{b} {\mathrm e}^{a x}-x^{b} \left (-a \right )^{b} b \left (-a x \right )^{-b} \Gamma \left (b , -a x \right )\right )}{a}+\frac {\left (-a \right )^{-b} c \left (x^{b} \left (-a \right )^{b} b \Gamma \left (b \right ) \left (-a x \right )^{-b}-x^{b} \left (-a \right )^{b} {\mathrm e}^{a x}-x^{b} \left (-a \right )^{b} b \left (-a x \right )^{-b} \Gamma \left (b , -a x \right )\right )}{a}\right )+c_{1} x^{-b} {\mathrm e}^{-a x} \end {align*}
which simplifies to \begin {align*} p \left (x \right ) &= \frac {{\mathrm e}^{-a x} \left (\left (b^{3}+3 b^{2}+2 b \right ) \Gamma \left (b , -a x \right )-\Gamma \left (b +3\right )\right ) c^{2} \left (-a x \right )^{-b}+x^{-b} c_{1} a^{3} {\mathrm e}^{-a x}-\left (\left (-a^{2} x^{2}+x \left (2+b \right ) a -b^{2}-3 b -2\right ) c +x \,a^{3}\right ) c}{a^{3}} \end {align*}
Since \(p=y^{\prime }\) then the new first order ode to solve is \begin {align*} y^{\prime } = \frac {{\mathrm e}^{-a x} \left (\left (b^{3}+3 b^{2}+2 b \right ) \Gamma \left (b , -a x \right )-\Gamma \left (b +3\right )\right ) c^{2} \left (-a x \right )^{-b}+x^{-b} c_{1} a^{3} {\mathrm e}^{-a x}-\left (\left (-a^{2} x^{2}+x \left (2+b \right ) a -b^{2}-3 b -2\right ) c +x \,a^{3}\right ) c}{a^{3}} \end {align*}
Integrating both sides gives \begin {align*} y = \int \frac {\left (-a x \right )^{-b} b^{3} c^{2} \Gamma \left (b , -a x \right ) {\mathrm e}^{-a x}+a^{2} c^{2} x^{2}+3 \left (-a x \right )^{-b} b^{2} c^{2} \Gamma \left (b , -a x \right ) {\mathrm e}^{-a x}-a^{3} x c -a \,c^{2} b x +x^{-b} c_{1} a^{3} {\mathrm e}^{-a x}+2 b \,c^{2} \left (-a x \right )^{-b} \Gamma \left (b , -a x \right ) {\mathrm e}^{-a x}-2 a x \,c^{2}+c^{2} b^{2}-c^{2} \left (-a x \right )^{-b} \Gamma \left (b +3\right ) {\mathrm e}^{-a x}+3 c^{2} b +2 c^{2}}{a^{3}}d x +c_{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \int \frac {\left (-a x \right )^{-b} b^{3} c^{2} \Gamma \left (b , -a x \right ) {\mathrm e}^{-a x}+a^{2} c^{2} x^{2}+3 \left (-a x \right )^{-b} b^{2} c^{2} \Gamma \left (b , -a x \right ) {\mathrm e}^{-a x}-a^{3} x c -a \,c^{2} b x +x^{-b} c_{1} a^{3} {\mathrm e}^{-a x}+2 b \,c^{2} \left (-a x \right )^{-b} \Gamma \left (b , -a x \right ) {\mathrm e}^{-a x}-2 a x \,c^{2}+c^{2} b^{2}-c^{2} \left (-a x \right )^{-b} \Gamma \left (b +3\right ) {\mathrm e}^{-a x}+3 c^{2} b +2 c^{2}}{a^{3}}d x +c_{2} \\ \end{align*}
Verification of solutions
\[ y = \int \frac {\left (-a x \right )^{-b} b^{3} c^{2} \Gamma \left (b , -a x \right ) {\mathrm e}^{-a x}+a^{2} c^{2} x^{2}+3 \left (-a x \right )^{-b} b^{2} c^{2} \Gamma \left (b , -a x \right ) {\mathrm e}^{-a x}-a^{3} x c -a \,c^{2} b x +x^{-b} c_{1} a^{3} {\mathrm e}^{-a x}+2 b \,c^{2} \left (-a x \right )^{-b} \Gamma \left (b , -a x \right ) {\mathrm e}^{-a x}-2 a x \,c^{2}+c^{2} b^{2}-c^{2} \left (-a x \right )^{-b} \Gamma \left (b +3\right ) {\mathrm e}^{-a x}+3 c^{2} b +2 c^{2}}{a^{3}}d x +c_{2} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x \left (\frac {d}{d x}y^{\prime }\right )+\left (a x +b \right ) y^{\prime }=-c x \left (-c \,x^{2}+a x +b +1\right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d}{d x}y^{\prime } \\ \bullet & {} & \textrm {Make substitution}\hspace {3pt} u =y^{\prime }\hspace {3pt}\textrm {to reduce order of ODE}\hspace {3pt} \\ {} & {} & x u^{\prime }\left (x \right )+\left (a x +b \right ) u \left (x \right )=-c x \left (-c \,x^{2}+a x +b +1\right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & u^{\prime }\left (x \right )=-\frac {\left (a x +b \right ) u \left (x \right )+c x \left (-c \,x^{2}+a x +b +1\right )}{x} \\ \bullet & {} & \textrm {Collect w.r.t.}\hspace {3pt} u \left (x \right )\hspace {3pt}\textrm {and simplify}\hspace {3pt} \\ {} & {} & u^{\prime }\left (x \right )=-\frac {\left (a x +b \right ) u \left (x \right )}{x}-c \left (-c \,x^{2}+a x +b +1\right ) \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} u \left (x \right )\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE}\hspace {3pt} \\ {} & {} & u^{\prime }\left (x \right )+\frac {\left (a x +b \right ) u \left (x \right )}{x}=-c \left (-c \,x^{2}+a x +b +1\right ) \\ \bullet & {} & \textrm {The ODE is linear; multiply by an integrating factor}\hspace {3pt} \mu \left (x \right ) \\ {} & {} & \mu \left (x \right ) \left (u^{\prime }\left (x \right )+\frac {\left (a x +b \right ) u \left (x \right )}{x}\right )=-\mu \left (x \right ) c \left (-c \,x^{2}+a x +b +1\right ) \\ \bullet & {} & \textrm {Assume the lhs of the ODE is the total derivative}\hspace {3pt} \frac {d}{d x}\left (u \left (x \right ) \mu \left (x \right )\right ) \\ {} & {} & \mu \left (x \right ) \left (u^{\prime }\left (x \right )+\frac {\left (a x +b \right ) u \left (x \right )}{x}\right )=u^{\prime }\left (x \right ) \mu \left (x \right )+u \left (x \right ) \mu ^{\prime }\left (x \right ) \\ \bullet & {} & \textrm {Isolate}\hspace {3pt} \mu ^{\prime }\left (x \right ) \\ {} & {} & \mu ^{\prime }\left (x \right )=\frac {\mu \left (x \right ) \left (a x +b \right )}{x} \\ \bullet & {} & \textrm {Solve to find the integrating factor}\hspace {3pt} \\ {} & {} & \mu \left (x \right )=x^{b} {\mathrm e}^{a x} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \left (\frac {d}{d x}\left (u \left (x \right ) \mu \left (x \right )\right )\right )d x =\int -\mu \left (x \right ) c \left (-c \,x^{2}+a x +b +1\right )d x +c_{1} \\ \bullet & {} & \textrm {Evaluate the integral on the lhs}\hspace {3pt} \\ {} & {} & u \left (x \right ) \mu \left (x \right )=\int -\mu \left (x \right ) c \left (-c \,x^{2}+a x +b +1\right )d x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} u \left (x \right ) \\ {} & {} & u \left (x \right )=\frac {\int -\mu \left (x \right ) c \left (-c \,x^{2}+a x +b +1\right )d x +c_{1}}{\mu \left (x \right )} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} \mu \left (x \right )=x^{b} {\mathrm e}^{a x} \\ {} & {} & u \left (x \right )=\frac {\int -x^{b} {\mathrm e}^{a x} c \left (-c \,x^{2}+a x +b +1\right )d x +c_{1}}{x^{b} {\mathrm e}^{a x}} \\ \bullet & {} & \textrm {Evaluate the integrals on the rhs}\hspace {3pt} \\ {} & {} & u \left (x \right )=\frac {-\frac {\left (-a \right )^{-b} c^{2} \left (x^{b} \left (-a \right )^{b} b \left (b^{2}+3 b +2\right ) \Gamma \left (b \right ) \left (-a x \right )^{-b}-x^{b} \left (-a \right )^{b} \left (a^{2} x^{2}-a b x -2 a x +b^{2}+3 b +2\right ) {\mathrm e}^{a x}-x^{b} \left (-a \right )^{b} b \left (b^{2}+3 b +2\right ) \left (-a x \right )^{-b} \Gamma \left (b , -a x \right )\right )}{a^{3}}-\frac {\left (-a \right )^{-b} c \left (x^{b} \left (-a \right )^{b} \left (1+b \right ) b \Gamma \left (b \right ) \left (-a x \right )^{-b}+x^{b} \left (-a \right )^{b} \left (a x -b -1\right ) {\mathrm e}^{a x}-x^{b} \left (-a \right )^{b} \left (1+b \right ) b \left (-a x \right )^{-b} \Gamma \left (b , -a x \right )\right )}{a}+\frac {\left (-a \right )^{-b} c b \left (x^{b} \left (-a \right )^{b} b \Gamma \left (b \right ) \left (-a x \right )^{-b}-x^{b} \left (-a \right )^{b} {\mathrm e}^{a x}-x^{b} \left (-a \right )^{b} b \left (-a x \right )^{-b} \Gamma \left (b , -a x \right )\right )}{a}+\frac {\left (-a \right )^{-b} c \left (x^{b} \left (-a \right )^{b} b \Gamma \left (b \right ) \left (-a x \right )^{-b}-x^{b} \left (-a \right )^{b} {\mathrm e}^{a x}-x^{b} \left (-a \right )^{b} b \left (-a x \right )^{-b} \Gamma \left (b , -a x \right )\right )}{a}+c_{1}}{x^{b} {\mathrm e}^{a x}} \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & u \left (x \right )=\frac {{\mathrm e}^{-a x} \left (\left (b^{3}+3 b^{2}+2 b \right ) \Gamma \left (b , -a x \right )-\Gamma \left (b +3\right )\right ) c^{2} \left (-a x \right )^{-b}+x^{-b} c_{1} a^{3} {\mathrm e}^{-a x}-\left (\left (-a^{2} x^{2}+x \left (2+b \right ) a -b^{2}-3 b -2\right ) c +x \,a^{3}\right ) c}{a^{3}} \\ \bullet & {} & \textrm {Solve 1st ODE for}\hspace {3pt} u \left (x \right ) \\ {} & {} & u \left (x \right )=\frac {{\mathrm e}^{-a x} \left (\left (b^{3}+3 b^{2}+2 b \right ) \Gamma \left (b , -a x \right )-\Gamma \left (b +3\right )\right ) c^{2} \left (-a x \right )^{-b}+x^{-b} c_{1} a^{3} {\mathrm e}^{-a x}-\left (\left (-a^{2} x^{2}+x \left (2+b \right ) a -b^{2}-3 b -2\right ) c +x \,a^{3}\right ) c}{a^{3}} \\ \bullet & {} & \textrm {Make substitution}\hspace {3pt} u =y^{\prime } \\ {} & {} & y^{\prime }=\frac {{\mathrm e}^{-a x} \left (\left (b^{3}+3 b^{2}+2 b \right ) \Gamma \left (b , -a x \right )-\Gamma \left (b +3\right )\right ) c^{2} \left (-a x \right )^{-b}+x^{-b} c_{1} a^{3} {\mathrm e}^{-a x}-\left (\left (-a^{2} x^{2}+x \left (2+b \right ) a -b^{2}-3 b -2\right ) c +x \,a^{3}\right ) c}{a^{3}} \\ \bullet & {} & \textrm {Integrate both sides to solve for}\hspace {3pt} y \\ {} & {} & \int y^{\prime }d x =\int \frac {{\mathrm e}^{-a x} \left (\left (b^{3}+3 b^{2}+2 b \right ) \Gamma \left (b , -a x \right )-\Gamma \left (b +3\right )\right ) c^{2} \left (-a x \right )^{-b}+x^{-b} c_{1} a^{3} {\mathrm e}^{-a x}-\left (\left (-a^{2} x^{2}+x \left (2+b \right ) a -b^{2}-3 b -2\right ) c +x \,a^{3}\right ) c}{a^{3}}d x +c_{2} \\ \bullet & {} & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y=\int \frac {{\mathrm e}^{-a x} \left (\left (b^{3}+3 b^{2}+2 b \right ) \Gamma \left (b , -a x \right )-\Gamma \left (b +3\right )\right ) c^{2} \left (-a x \right )^{-b}+x^{-b} c_{1} a^{3} {\mathrm e}^{-a x}-\left (\left (-a^{2} x^{2}+x \left (2+b \right ) a -b^{2}-3 b -2\right ) c +x \,a^{3}\right ) c}{a^{3}}d x +c_{2} \end {array} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable -> Calling odsolve with the ODE`, diff(_b(_a), _a) = -(-c^2*_a^3+c*_a^2*a+_a*_b(_a)*a+_a*b*c+_b(_a)*b+c*_a)/_a, _b(_a)` *** Sublev Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful <- high order exact linear fully integrable successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 115
dsolve(x*diff(y(x),x$2)+(a*x+b)*diff(y(x),x)+c*x*(-c*x^2+a*x+b+1)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{2} a^{3}-\left (\int \left (-c^{2} \left (\left (b^{3}+3 b^{2}+2 b \right ) \Gamma \left (b , -a x \right )-\Gamma \left (b +3\right )\right ) {\mathrm e}^{-a x} \left (-a x \right )^{-b}-{\mathrm e}^{-a x} x^{-b} c_{1} a^{3}+\left (\left (-b^{2}+\left (a x -3\right ) b -a^{2} x^{2}+2 a x -2\right ) c +a^{3} x \right ) c \right )d x \right )}{a^{3}} \]
✓ Solution by Mathematica
Time used: 61.322 (sec). Leaf size: 92
DSolve[x*y''[x]+(a*x+b)*y'[x]+c*x*(-c*x^2+a*x+b+1)==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \int _1^xe^{-a K[1]} K[1]^{-b} \left (\frac {c \left (-\left ((b+1) \Gamma (b+1,-a K[1]) a^2\right )+\Gamma (b+2,-a K[1]) a^2+c \Gamma (b+3,-a K[1])\right ) K[1]^b (-a K[1])^{-b}}{a^3}+c_1\right )dK[1]+c_2 \]